Irreducibility and normality conjecture for nilpotent commuting varieties of type A

Let $G$ be a reductive algebraic group of type $A$ with Lie algebra $\mathfrak{g}$, let $B$ be a Borel subgroup with nilpotent radical $\mathfrak{u}$, and let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$. For $r\geq 1$, write $C_r(\mathcal{N})$ for the variety of pairwise commuting $r$-tuples in $\mathcal{N}$. Type A nilpotent commuting-variety conjecture. The variety $C_r(\mathcal{N})$ is irreducible and normal; equivalently, the morphism

$$m:G\times^B C_r(\mathfrak{u})\to C_r(\mathcal{N})$$

satisfies all the hypotheses of Zariski's Main Theorem. The source proves the assertion for types $A_1$ and $A_2$, while the result for type $A_n$ with arbitrary $n$ and $r>2$ remains open.